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Question:

Show that $f(x)=x+\cos x-a$ is an increasing function on $R$ for all values of a ?

Solution:

We have,

$f(x)=x+\cos x-a$

$f^{\prime}(x)=1-\sin x=\frac{2 \cos ^{2} x}{2}$

Now,

$x \in R$

$\Rightarrow \frac{\cos ^{2} x}{2}>0$

$\Rightarrow \frac{2 \cos ^{2} x}{2}>0$

$\Rightarrow f^{\prime}(x)>0$

Hence, $f(x)$ is an increasing function for $x \in R$

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